fix(projects): tighten public project listing filters

[vcnainala]

Summary

• Exclude deleted projects from the public projects page and public text search
• Only show public projects that have at least one public, non-archived, non-deleted study
• Remove redundant notification prop from AppLayout (component already reads from page props)

Commits

1. fix(projects): only list public projects with visible studies
2. fix(ui): stop passing stale notifications prop in AppLayout

Test plan

• php artisan test tests/Feature/PublicProjectsPageTest.php
• Visit /projects and confirm only projects with visible public studies appear
• Confirm notifications dropdown still works in the app layout

[codecov]

Codecov Report

✅ All modified and coverable lines are covered by tests.
✅ Project coverage is 76.62%. Comparing base (f9fb6af) to head (32e3775).

Additional details and impacted files

@@                Coverage Diff                @@
##             development    #1458      +/-   ##
=================================================
+ Coverage          76.59%   76.62%   +0.03%     
  Complexity          3590     3590              
=================================================
  Files                249      249              
  Lines              13510    13530      +20     
=================================================
+ Hits               10348    10368      +20     
  Misses              3162     3162              

Flag	Coverage Δ
unittests	76.62% <100.00%> (+0.03%)	⬆️

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